Find dy/dx y=x^12

Solution:

Step 1:

Apply the differentiation operator $\frac{d}{dx}$ to both sides of the equation $y = x^{12}$.

Step 2:

The derivative of $y$ in terms of $x$ is denoted as $\frac{dy}{dx}$.

Step 3:

Utilize the Power Rule for differentiation, which asserts that the derivative of $x^n$ is $nx^{n-1}$, where in this case $n$ equals 12, to find the derivative of $x^{12}$.

Step 4:

Combine the results to form the new equation by equating the derivative of $y$ to the derivative of $x^{12}$.

Step 5:

Substitute $\frac{dy}{dx}$ for $y$ to express the final derivative of the function.

Knowledge Notes:

The process of finding the derivative of a function is a fundamental concept in calculus, known as differentiation. The derivative represents the rate at which a function is changing at any given point and is a cornerstone of differential calculus.

The Power Rule is a basic differentiation rule used to find the derivative of a function of the form $x^n$, where $n$ is any real number. According to the Power Rule, the derivative of $x^n$ is $nx^{n-1}$. This rule simplifies the process of differentiation when dealing with polynomial functions.

In the given problem, we are asked to find the derivative of $y$ with respect to $x$, where $y$ is a function of $x$ defined as $y = x^{12}$. The differentiation process involves the following steps:

1. Apply the differentiation operator $\frac{d}{dx}$ to both sides of the function equation.

2. Recognize that the derivative of $y$ with respect to $x$ is written as $\frac{dy}{dx}$.

3. Use the Power Rule to differentiate $x^{12}$, which results in $12x^{11}$.

4. Write the derivative of $y$ as equal to the derivative of $x^{12}$, forming the equation $\frac{dy}{dx} = 12x^{11}$.

5. Recognize that $\frac{dy}{dx}$ is the final derivative of the function $y = x^{12}$ with respect to $x$.

Understanding the Power Rule and its application is essential for solving this problem and is a key skill in calculus.